## Abstract

Optical gradient force in a parity-time (PT)-symmetric coupled-waveguide system is theoretically studied. We find that when the system evolves from PT-symmetric region to broken-PT-symmetric region, the normalized optical forces of the two eigenmodes decrease first and become the same when the exceptional point is reached. Besides, the optical force induced PT phase transition is demonstrated. It is worth noting that, when the system is in the broken-PT-symmetric region and the length of the waveguide is much longer than the propagation length of the lossy eigenmode, the total optical gradient force acting on the two waveguides will decrease with the decreasing of the gap. This work gives us a new understanding of integrated optomechanics by combining with PT symmetry.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Hamiltonians should be Hermitian, which is one of the basic postulates of quantum mechanics. Hermiticity guarantees that the energy spectrum is real and the time evolution of a quantum system is unitary [1]. However, in 1998, Bender et al. showed that non-Hermitian Hamiltonians can still exhibit entirely real spectra, as long as the Hamiltonians obey the principle of PT symmetry [2]. A Hamiltonian associated with a complex potential $V(\widehat{x})$possesses PT symmetry provided that it commutes with$\widehat{P}\widehat{T}$ operator, i.e., $[\widehat{P}\widehat{T},\widehat{H}]=0$. In which cases, one can show that the complex potential must satisfy the necessary condition of$V(\widehat{x})={V}^{\ast}(-\widehat{x})$. In other words, the real part of the complex potential V(x) is an even function of the position, whereas the imaginary part is an odd function. An extremely interesting phenomenon associated with PT-symmetric Hamiltonians is spontaneous PT-symmetry breaking. Once it crosses a certain phase transition point, the system loses its PT symmetry and its spectra become complex [3, 4]. The study on PT symmetry is not just limited to quantum mechanics [5] but also has spread into many other branches of physics, such as the microwave and acoustic sensor realms [6, 7]. Due to the formal equivalence of the optical wave equation and the Schrödinger equation, it makes optical systems a fertile ground to study PT symmetry [8, 9]. In optical PT-symmetric systems, the complex permittivity *ε*(**r**) plays the role of the potential, and the necessary condition of the complex potential $V(\widehat{x})$amounts to *ε*(**r**) = ${\epsilon}^{*}$(**r**) [10]. Several interesting properties can exhibit in PT-symmetric optical arrangements, such as unidirectional invisibility [11] and nonreciprocity of light propagation [12,13]. When periodicity is introduced, other exciting phenomena can be observed in PT symmetric structures, such as Bloch oscillations [14] and solitons [15]. Likewise, important applications of PT symmetry such as single-mode PT-symmetric lasers [16,17] and unidirectional reflectionless PT-symmetric metamaterial at optical frequencies [18] have also emerged.

In parallel to works above, optomechanics has raised a great deal of attention in recent years [19, 20]. Ultrasmall-mass and ultrahigh-mechanical-frequency optomechanical systems and strong optical gradient force are particularly desirable [21–24]. The optical gradient force has been exploited for actuating optical waveguides [25]. When two closely placed waveguides are freestanding, this interaction force can cause nanometer or even micrometer-level mechanical displacements of the waveguides. Due to the displacement, an enhanced Kerr coefficient, namely mechanical Kerr coefficient, which is several orders of magnitude larger than the intrinsic optical Kerr coefficient, is obtained [26]. Optical gradient forces have been associated with many remarkable phenomena such as optomechanical sensing [27], synchronization of nanomechanical oscillators [28], optical nonreciprocity [29–31], and optomechanical single-photon frequency shift [32].

In this paper, we study optical gradient forces in PT-symmetric coupled-waveguide systems. We calculate the normalized optical forces *f _{n}* (nN/μm/mW) of the two eigenmodes, i.e, the optical forces per unit length normalized to the local power, when the system evolves from PT-symmetric region to broken-PT-symmetric region. Besides, we show that the optical force can induce PT phase transition. When the system in the broken-PT-symmetric region and the length of the waveguide is much longer than the propagation length of the lossy eigenmode, the total optical force acting on the two waveguides decrease as the gap between the two waveguides decreases. All of these properties are significantly different from those in conventional Hermitian coupled-waveguide systems where the gain/loss constant$\gamma =0$.

## 2. Device structure and the characteristics of the optical force

The structure of a PT-symmetric coupler consisting of two suspended and double-clamped beams (DCB) is shown in Fig. 1. Two coupled waveguides have balanced gain and loss coefficient, i.e., ${\gamma}_{g}={\gamma}_{l}=\gamma $, and the coupling coefficient between the two waveguides is *κ* [6,8]. In this work we set the material of the DCB is silicon nitride (Si_{3}N_{4}) with the real part of the relative permittivity ${\epsilon}_{r}=4$ and the substrate is silica. Both waveguide cross sections are 750 nm $\times $ 400 nm with the length *L* = 300 μm. The optical wavelength is 1550 nm. Optical gradient force between the two waveguides will induce deflection of the DCB in the y-z plane, and the silica groove is deep enough that the optical force between the waveguide and the silica substrate can be neglected.

When the system is in the PT-symmetric region ($\gamma <2\kappa $), two eigenmodes are given by $|1,2\u3009=(1,\pm {e}^{\pm i\theta})$, with corresponding eigenvalues $\pm \mathrm{cos}\theta $ and $\mathrm{sin}\theta =\gamma /2\kappa $. When PT symmetry is broken($\gamma >2\kappa $), the eigenmodes become $|1,2\u3009=(1,i{e}^{\mp \theta})$, and corresponding eigenvalues are $\mp isinh\theta $ with $\mathrm{cosh}\theta =\gamma /2\kappa $. As we know that the gain/loss constant *γ* is related to the imaginary part of the relative permittivity ${\epsilon}_{i}$ and the mode field distribution. When ${\epsilon}_{i}=0$, there is no material gain and loss and the eigenmodes become conventional symmetric and antisymmetric supermodes, i.e. $|S,AS\u3009=(1,\pm 1)$ .

In Fig. 2(a) we fix the gap between the waveguides at 450 nm and change the gain/loss constant. We can see that when crossing the exceptional point corresponding to ${\epsilon}_{i}=0.0366$, PT symmetry is broken and the real part of the effective refractive indices of the two eigenmodes are the same while the imaginary parts of the effective refractive indices are opposite, which means one of the two supermodes will experience gain while the other will experience loss. Meanwhile, the normalized optical gradient force is obtained by integrating the time-averaged component of the Maxwell stress tensor $\overleftrightarrow{T}$ by [33]

whereHere, we give a qualitative explanation of the evolution of the normalized optical force with increasing ${\epsilon}_{i}$ as shown in Fig. 2(b). As we know that the optical force can be repulsive or attractive depending on the relative phase of individual guided modes in the two waveguides [34–36]. When ${\epsilon}_{i}=0$, for the symmetric supermode, the phase difference ${\phi}_{0}$between field components in the two waveguides is 0, so the fields in the two waveguides are in-phase at every moment as shown in Fig. 3(a). i.e., $|S\u3009=({e}^{i\omega t},{e}^{i\omega t})$. So the optical force is always attractive in an electromagnetic oscillation period. For the antisymmetric supermode, the phase difference is always π, and the eigenmode becomes $|AS\u3009=({e}^{i\omega t},{e}^{i\omega t+\pi})$. So the field components in the two waveguides are always reversed-phase, as shown in Fig. 3(b), and the optical force is repulsive at all the time in an electromagnetic oscillation period. However, when ${\epsilon}_{i}\ne 0$ and the system is in the PT-symmetric region, the phase difference ${\phi}_{0}$ between the field components of the eigenmodes is neither 0 nor π. The eigenmodes become $|1,2\u3009=({e}^{i\omega t},{e}^{i(\omega t\pm {\phi}_{0})})$, and this means the optical field seems like a symmetric mode in some time while like an antisymmetric mode in the other time, as is shown in Fig. 3(c). So the sign of the optical force will change in an oscillation period. Consequently, the magnitudes of net time-averaged optical forces of the two eigenmodes are smaller than those when ${\epsilon}_{i}=0$. Until the exceptional point is reached, where $|1\u3009$ and $|2\u3009$ are degenerate and thus the normalized optical forces *f _{n}* of the two eigenmodes are equal to each other, as shown in Fig. 2 (b).

As we know, by using nonlinear effects, the refractive index of the waveguide can be changed and thus the coupling coefficient *κ* will be changed too. So optical nonlinearity can be used to induce PT phase transition [37,38]. However, *κ* is also a function of the gap between the two waveguides. As is shown in Figs. 2(c) and 2(d), when ${\epsilon}_{i}$ is fixed at 0.0368, we can also get the exceptional point by changing the gap.

It can be see clearly in Figs. 4(a) and 4(b) that the mode field distribution is symmetric with respect to the y = 0 plane when the system is below the exceptional point corresponding to g_{0} = 448 nm, so the eigenmodes do not experience net gain or loss and the eigenvalue spectrum is real. However, as depicted in Figs. 4(c) and 4(d), once the gap is larger than g_{0}, $\gamma >2\kappa $, PT symmetry is spontaneously broken that one of the two eigenmodes predominantly locating in the gain waveguide and the other in the lossy one and the mode field distributions of the eigenmodes are centrosymmetric with each other. Because the net optical force of the system in y direction for each eigenmodes is zero, combining with the symmetry properties of the mode fields, which means the normalized optical forces of the two eigenmodes are still equal to each other in broken-PT-symmetric region. Besides, the symmetry properties of the mode fields of the eigenmodes also indicate that the real parts of the effective refractive indices are equal but the imaginary parts of the effective refractive indices are opposite of each other as shown in Figs. 2(a)-2(d), which means that after the light travel a distance about *L _{m}* = 1/(2|Im(β)|), where the power is

*e*

^{−1}of the initial power, the power of the lossy mode can be neglected and only the gain dominated eigenmode can exist.

By contrast, when ${\epsilon}_{i}=0$, symmetric and antisymmetric supermodes are always orthogonal. Whatever the gap is, the two eigenmodes can never be degenerate and corresponding the real part of *n _{eff}* and the normalized optical forces will never merge together, as shown in Figs. 2(e) and 2(f). Therefore, the relation between the normalized optical force and the waveguide gap in the PT symmetric coupler is very different from that in the conventional coupler.

## 3. Optical force induced PT phase transition

As mentioned above, we can realize the PT phase transition by changing the gap. Here we use the optical orce to change the gap in the PT-symmetric coupler structure. To describe the optical force induced deflection we must obtain the optical force distribution along the waveguide and thus the light field distribution along the waveguides. We present the analysis based on a complex coupled mode theory model [39], where the gain or loss induced permittivity perturbation Δ*ε*(x,y) is defined as the difference between the permittivity profile of the PT-symmetric coupler structure $\tilde{\epsilon}$ (x,y) and that of the conventional Hermitian coupler structure with *γ* = 0, i.e., *ε*(x,y). So the permittivity distribution can be expressed as $\tilde{\epsilon}$ (x,y) = *ε*(x,y) + Δ*ε*(x,y), and the coupled equation of the two eigenmodes is:

*E*denotes the transverse component of the electric field

_{tm}*E*. The refractive index is used to replace the permittivity function, i.e.,$\epsilon ={\epsilon}_{0}{n}^{2}$ and ${N}_{m}=\frac{1}{2}{\displaystyle \iint {E}_{tm}\times {E}_{tm}\cdot ds}$. ${\gamma}_{k}$represents the Kerr nonlinear coefficient. It is important to note that

_{m}*a*here is the eigenmode amplitude of the Hermitian coupler structure without gain and loss, i.e., the symmetric or antisymmetric supermode, and

_{m}*β*is the corresponding propagation constant. As the two eigenmodes are always excited simultaneously, we expand the total optical field by the eigenmodes of the Hermitian coupler structure with:where

_{m}*φ*(z) and

_{s,as}*E*are the phase and the mode field of the eigenmodes respectively. By substituting Eq. (5) into the Maxwell stress tensor $\overleftrightarrow{T}$ in Eqs. (1) and (2), the result can be divided into three parts, which is given by [32]where

_{s,as}*F*(z) is the time-averaged total optical force per unit length acting on the two waveguides. The first two terms on the right-hand side are the forces resulting from the symmetric and anti-symmetric modes respectively, and the third term is the optical force resulting from the beating between these two eigenmodes.

*f*and

_{n,s}*f*are shown in Fig. 2(f).

_{n,as}*f*is related to the incidence condition and it can also be calculated by the finite-element method. By using Eqs. (3) and (4), we can obtain the eigenmode amplitude distribution along the waveguides

_{n,b}*a*(z). Combing

_{m}*a*(z) with the

_{m}*f*,

_{n,s}*f*and

_{n,as}*f*,

_{n,b}*F*(z) can be calculated as well, and the optical force induced deflection

*u*(

*z*) of the DCB is given by:where

*Y*is the Young’s Modulus,

*I*is the cross sectional area moment of inertia with respect to the neutral axis.

When an initial optical field distribution along the DCB is given, we can calculate the optical force distribution and the optical force induced waveguide deflection. The waveguide deflection will, in turn, change the optical field distribution, and this will give us a new optical force distribution. Hence, according to the connection among the optical field distribution, the optical force and the optical force induced deflection, an iterative cycle of numerical calculations can be established by using Eqs. (1)-(7) and a self-consistent steady-state solution of the optical force and the waveguide deflection can be obtained finally.

Figure 5(a) depicts the optical force induced deflection at different input powers with symmetric mode incident and *ε _{i}* fixed at 0.0368. As we know, the mechanical Kerr coefficient is several orders of magnitude larger than the optical Kerr coefficient [26], and the latter of silicon nitride is one order of magnitude smaller than that of silicon [40, 41], so the mechanical Kerr coefficient is much larger than the optical Kerr coefficient in Si

_{3}N

_{4}waveguides. We can see from Fig. 5(a) that the waveguide deflection is almost unchanged when the optical Kerr nonlinearity is considered. This means that although the effective refractive index change induced by the optical Kerr nonlinearity may be large when the optical power is high, it is still much smaller than that induced by the mechanical Kerr nonlinearity, i.e., the effective refractive index changing induced by gap changed. When the input power is smaller than 1.5 mW, the minimum gap is larger than g

_{0}= 448 nm and thus PT symmetry is broken at every position along the DCB [shown in Fig. 2(c)]. When the input power is larger than 1.5 mW, the minimum gap is smaller than g

_{0}. The DCB will be divided into three regions along the z direction, that is, the broken-PT-symmetric region, the PT-symmetric region, and the broken-PT-symmetric region, respectively.

It can be found more clearly in Fig. 5(b) that when the input power is smaller than 1.5 mW, the normalized optical forces of the two eigenmodes are equal at every position along the DCB. However, when the input power is larger than 1.5 mW, the normalized optical forces of the two eigenmodes are different from each other in the PT-symmetric region. This means by using the optical force, we can realize the PT phase transition between PT symmetry and broken PT symmetry.

In Fig. 5(c), to confirm the optical force induced PT phase transition we measure the output powers of the two waveguides as a function of the input power. As we can see from Fig. 2(c) that when the gap is larger than g_{0,} the effective gain/loss constant ${\gamma}_{eff}=\left|\mathrm{Im}(\beta )\right|=\sqrt{{(\gamma /2)}^{2}-{\left|\kappa \right|}^{2}}$ of the eigenmodes gets smaller with gap decreased [6], and there is no effective gain when the gap is smaller than g_{0}, i.e., ${\gamma}_{eff}=0$.

With the input power increased at first, due to the nonzero effective gain/loss constant, the propagation power grows exponentially and the deflection becomes larger. Once the minimum gap is smaller than g_{0}, the exponentially growing of the propagation power will be terminated in the PT-symmetric region. Until the light passes through the PT-symmetric region, the propagation power can exponentially grow again. Further increasing the input power will enlarge the length of the PT-symmetric region. So the average gain constant ${\gamma}_{avg}=({\displaystyle {\int}_{0}^{L}{\gamma}_{eff}dz})/L$ gets smaller when the input power is increased, where *L* is the length of the DCB. Consequently, the output power ${P}_{out}\propto {P}_{in}\cdot {e}^{{\gamma}_{avg}\cdot L}$ will exhibit a non-Hermitian behavior as shown in Fig. 5(c), that is, with the input power increased, the output power grows first and decays subsequently. Before the maximum output power is reached, the influence of the input power increment is dominant, and after that the influence of the average gain constant decrement is dominant.

## 4. Total optical gradient force

In Fig. 6 we calculate the total optical force acting on waveguides normalized to the input power, *F _{n}*, versus waveguides gap, which can be obtained by:

*F*(

*z*) is calculated by Eq. (6).

As we know, *F _{n}* is related to the initial phase difference between the incident light fields at the two waveguides, and both the two eigenmodes are always excited simultaneously. When g>g

_{0}, the system is in the broken-PT-symmetric region. As long as

*L*is much larger than

*L*, the loss dominated eigenmode will decay in a short distance and the gain dominated eigenmode will propagate with the exponentially increasing of the power. So the power in the two waveguides can still be approximatively regarded as exponentially growing. However, just as mentioned above, the effective gain coefficient decreases with the gap decreased in the broken-PT-symmetric region, which makes the power in the system reduce with gap decreased when the input power is fixed. So, the smaller gap corresponds to the smaller total optical force in the broken-PT-symmetric region as we can see from Fig. 6(a). But when g<g

_{m}_{0}, system is in the PT-symmetric region, there is no gain and loss and the sign of the normalized optical forces of the two eigenmodes are opposite. So the situation is much more complicated when both of the two eigenmodes are exist. As we can see from the inset of Fig. 6(a) that when initial phase differences of incident light ${\phi}_{0}=\pi $, the relationship between$\left|{F}_{n}\right|$ and gap are non-monotonic. The sign of

*F*can change from positive to negative when gap is changed.

_{n}In the conventional Hermitian coupler structure, the normalized optical force of the two eigenmodes are exponentially growing when the gap decreases as shown in Fig. 2(f), so when only one eigenmode is excited, the total optical force grows with the decreasing of the gap as we can see from Fig. 6(b). But similar to that in the PT-symmetric region, when both of the two eigenmodes are excited and a proper initial phase differences ${\phi}_{0}=0.54\pi $ is chosen, the relationship between$\left|{F}_{n}\right|$ and gap will be non-monotonic, as shown in the inset of Fig. 6(b). So in the PT symmetric coupled-waveguide waveguides, once the *L* is much larger than *L _{m}* and the system is in the broken-PT-symmetric region, whatever the incident situation is, the total optical force is decreased with gap decreased, which is totally different from that in the conventional Hermitian coupler structure.

## 5. Conclusion

To conclude, we have studied the normalized optical force of the two eigenmodes when the system evolves from PT-symmetric region to broken-PT-symmetric region. We find that the normalized optical forces of the two eigenmodes will be the same when the exceptional point is reached. The optical force induced PT phase transition and thus a non-Hermitian evolution of the output powers versus the input power are also demonstrated. Besides, we find that when the system is in the broken-PT-symmetric region and the length of the DCB is much longer than *L _{m}*, the total optical force in the PT-symmetric coupler structure will decrease with the decreasing of the gap. All of these properties are different from those in the conventional Hermitian coupler structure. This work offers us a new platform and understanding of integrated optomechanics.

## Funding

National Natural Science Foundation of China (11774110, 61307075); National Key Research and Development Program of China (2016YFB0402503).

## References and links

**1. **C. M. Bender, “Making sense of non-Hermitian Hamiltonians,” Rep. Prog. Phys. **70**(6), 947–1018 (2007). [CrossRef]

**2. **C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. **80**(24), 5243–5246 (1998). [CrossRef]

**3. **L. Feng, R. El-Ganainy, and L. Ge, “Non-Hermitian photonics based on parity–time symmetry,” Nat. Photonics **11**(12), 752–762 (2017). [CrossRef]

**4. **R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nat. Phys. **14**(1), 11–19 (2018). [CrossRef]

**5. **C. M. Bender, D. C. Brody, and H. F. Jones, “Complex Extension of Quantum Mechanics,” Phys. Rev. Lett. **89**(27), 270401 (2002). [CrossRef] [PubMed]

**6. **S. Bittner, B. Dietz, U. Günther, H. L. Harney, M. Miski-Oglu, A. Richter, and F. Schäfer, “PT Symmetry and Spontaneous Symmetry Breaking in a Microwave Billiard,” Phys. Rev. Lett. **108**(2), 024101 (2012). [CrossRef] [PubMed]

**7. **R. Fleury, D. Sounas, and A. Alù, “An invisible acoustic sensor based on parity-time symmetry,” Nat. Commun. **6**(1), 5905 (2015). [CrossRef] [PubMed]

**8. **A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. **103**(9), 093902 (2009). [CrossRef] [PubMed]

**9. **A. A. Zyablovsky, A. P. Vinogradov, A. A. Pukhov, A. V. Dorofeenko, and A. A. Lisyansky, “PT-symmetry in optics,” Phys.- Usp. **57**(11), 1063–1082 (2014). [CrossRef]

**10. **C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity–time symmetry in optics,” Nat. Phys. **6**(3), 192–195 (2010). [CrossRef]

**11. **Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional Invisibility Induced by PT-Symmetric Periodic Structures,” Phys. Rev. Lett. **106**(21), 213901 (2011). [CrossRef] [PubMed]

**12. **B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. **10**(5), 394–398 (2014). [CrossRef]

**13. **L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics **8**(7), 524–529 (2014). [CrossRef]

**14. **S. Longhi, “Bloch Oscillations in Complex Crystals with PT Symmetry,” Phys. Rev. Lett. **103**(12), 123601 (2009). [CrossRef] [PubMed]

**15. **M. Wimmer, A. Regensburger, M. A. Miri, C. Bersch, D. N. Christodoulides, and U. Peschel, “Observation of optical solitons in PT-symmetric lattices,” Nat. Commun. **6**(1), 7782 (2015). [CrossRef] [PubMed]

**16. **L. Feng, Z. J. Wong, R. M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science **346**(6212), 972–975 (2014). [CrossRef] [PubMed]

**17. **H. Hodaei, M. A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science **346**(6212), 975–978 (2014). [CrossRef] [PubMed]

**18. **L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. **12**(2), 108–113 (2013). [CrossRef] [PubMed]

**19. **D. Van Thourhout and J. Roels, “Optomechanical device actuation through the optical gradient force,” Nat. Photonics **4**(4), 211–217 (2010). [CrossRef]

**20. **J. Ma and M. L. Povinelli, “Applications of optomechanical effects for on-chip manipulation of light signals,” Curr. Opin. Solid. St. M. **16**(2), 82–90 (2012). [CrossRef]

**21. **X. Sun, J. Zheng, M. Poot, C. W. Wong, and H. X. Tang, “Femtogram Doubly Clamped Nanomechanical Resonators Embedded in a High-Q Two-Dimensional Photonic Crystal Nanocavity,” Nano Lett. **12**(5), 2299–2305 (2012). [CrossRef] [PubMed]

**22. **H. Li, J. W. Noh, Y. Chen, and M. Li, “Enhanced optical forces in integrated hybrid plasmonic waveguides,” Opt. Express **21**(10), 11839–11851 (2013). [CrossRef] [PubMed]

**23. **Z. Huang, K. Cui, G. Bai, X. Feng, F. Liu, W. Zhang, and Y. Huang, “High-mechanical-frequency characteristics of optomechanical crystal cavity with coupling waveguide,” Sci. Rep. **6**(1), 34160 (2016). [CrossRef] [PubMed]

**24. **X. Xu, L. Shi, Y. Liu, Z. Wang, and X. Zhang, “Enhanced optical gradient forces between coupled graphene sheets,” Sci. Rep. **6**(1), 28568 (2016). [CrossRef] [PubMed]

**25. **M. L. Povinelli, M. Lončar, M. Ibanescu, E. J. Smythe, S. G. Johnson, F. Capasso, and J. D. Joannopoulos, “Evanescent-wave bonding between optical waveguides,” Opt. Lett. **30**(22), 3042–3044 (2005). [CrossRef] [PubMed]

**26. **J. Ma and M. L. Povinelli, “Mechanical Kerr nonlinearities due to bipolar optical forces between deformable silicon waveguides,” Opt. Express **19**(11), 10102–10110 (2011). [CrossRef] [PubMed]

**27. **Y. W. Hu, Y. F. Xiao, Y. C. Liu, and Q. Gong, “Optomechanical sensing with on-chip microcavities,” Front. Phys. **8**(5), 475–490 (2013). [CrossRef]

**28. **M. Bagheri, M. Poot, L. Fan, F. Marquardt, and H. X. Tang, “Photonic Cavity Synchronization of Nanomechanical Oscillators,” Phys. Rev. Lett. **111**(21), 213902 (2013). [CrossRef] [PubMed]

**29. **Z. Wang, L. Shi, Y. Liu, X. Xu, and X. Zhang, “Optical nonreciprocity in asymmetric optomechanical couplers,” Sci. Rep. **5**(1), 8657 (2015). [CrossRef] [PubMed]

**30. **K. Fang, J. Luo, A. Metelmann, M. H. Matheny, F. Marquardt, A. A. Clerk, and O. Painter, “Generalized non-reciprocity in an optomechanical circuit via synthetic magnetism and reservoir engineering,” Nat. Phys. **13**(5), 465–471 (2017). [CrossRef]

**31. **E. Verhagen and A. Alù, “Optomechanical nonreciprocity,” Nat. Phys. **13**(10), 922–924 (2017). [CrossRef]

**32. **L. Fan, C. L. Zou, M. Poot, R. Cheng, X. Guo, X. Han, and H. X. Tang, “Integrated optomechanical single-photon frequency shifter,” Nat. Photonics **10**(12), 766–770 (2016). [CrossRef]

**33. **J. D. Jackson, *Classical Electrodynamics,* 3rd ed. (Wiley, 1998).

**34. **M. Li, W. H. P. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature **456**(7221), 480–484 (2008). [CrossRef] [PubMed]

**35. **M. Li, W. H. P. Pernice, and H. X. Tang, “Tunable bipolar optical interactions between guided lightwaves,” Nat. Photonics **3**(8), 464–468 (2009). [CrossRef]

**36. **J. Roels, I. De Vlaminck, L. Lagae, B. Maes, D. Van Thourhout, and R. Baets, “Tunable optical forces between nanophotonic waveguides,” Nat. Nanotechnol. **4**(8), 510–513 (2009). [CrossRef] [PubMed]

**37. **Y. Lumer, Y. Plotnik, M. C. Rechtsman, and M. Segev, “Nonlinearly Induced PT transition in Photonic Systems,” Phys. Rev. Lett. **111**(26), 263901 (2013). [CrossRef] [PubMed]

**38. **W. Walasik, C. Ma, and N. M. Litchinitser, “Nonlinear parity-time-symmetric transition in finite-size optical couplers,” Opt. Lett. **40**(22), 5327–5330 (2015). [CrossRef] [PubMed]

**39. **W. P. Huang and J. Mu, “Complex coupled-mode theory for optical waveguides,” Opt. Express **17**(21), 19134–19152 (2009). [CrossRef] [PubMed]

**40. **K. Ikeda, R. E. Saperstein, N. Alic, and Y. Fainman, “Thermal and Kerr nonlinear properties of plasma-deposited silicon nitride/ silicon dioxide waveguides,” Opt. Express **16**(17), 12987–12994 (2008). [CrossRef] [PubMed]

**41. **C. J. Krückel, A. Fülöp, T. Klintberg, J. Bengtsson, P. A. Andrekson, and V. Torres-Company, “Linear and nonlinear characterization of low-stress high-confinement silicon-rich nitride waveguides,” Opt. Express **23**(20), 25827–25837 (2015). [CrossRef] [PubMed]